Phillips, P. C. B. Towards a unified asymptotic theory for autoregression. (English) Zbl 0654.62073 Biometrika 74, 535-547 (1987). Let \(y_ 1,...,y_ T\) be generated by the model \(y_ t=ay_{t-1}+u_ t\) where \(y_ 0\) is any given random variable, the innovations \(\{u_ t\}\) satisfy some rather general moment and weak dependence conditions and \(a=\exp (c/T)\) with \(-\infty <c<\infty\). If \(c\neq 0\) then \(\{y_ t\}\) is called near-integrated. The author derives the asymptotic distributions of \[ T^{-3/2}\sum y_ t,\quad T^{-2}\sum y\quad 2_ t,\quad T^{-1}\sum y_{t-1}u_ t\quad and\quad \hat a=\sum y_ ty_{t-1}/\sum y\quad 2_{t-1}\quad as\quad T\to \infty. \] The asymptotics of some expressions are analysed also for \(c\to \pm \infty\). The general theory is expressed in terms of functionals of a simple diffusion process. Reviewer: J.Anděl Cited in 2 ReviewsCited in 238 Documents MSC: 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 60F05 Central limit and other weak theorems Keywords:first-order autoregression; noncentrality parameter; unit root; asymptotic theory; continuous time estimation; asymptotic power of tests; local alternatives; Brownian motion; near-integrated process; moment conditions; weak dependence conditions; diffusion process PDFBibTeX XMLCite \textit{P. C. B. Phillips}, Biometrika 74, 535--547 (1987; Zbl 0654.62073) Full Text: DOI Link